In this section we study two types of examples which illustrate the
potential of the presented numerical approach. First we consider
initial data in the form of a mollified (smoothed out) step function. Then we study
initial data not satisfying the Faddeev condition
\(\int ^{+\infty }_{-\infty } (1 +|x|)|u(x)|dx <\infty \) , see
[1]}, but nevertheless vanishing for
\(|x|\rightarrow \infty \) . These examples are studied for two types of
nonlinearity, for \(p=2\) , the completely integrable KdV equation, and
for \(p=4\) , a non-integrable generalised KdV equation discussed for
instance in [2]}. The latter is
still sub-critical which means that there is no blow-up for
sufficiently regular initial data.
